Circumference of 3-connected cubic graphs

The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $\mathrm{\Omega }(n^{0.694})$. Bilinski {\it et al.} improved this lower bound to $\mathrm{\Omega }(n^{0.753})$ by studying large Eulerian subgraphs in 3-edge-connected graphs. In this paper, we further improve this lower bound to $\mathrm{\Omega }(n^{0.8})$. This is done by considering certain 2-connected cubic graphs, finding cycles through two given edges, and distinguishing the cases whether or not these edges are adjacent.